Using the free energy surface one can employ a probabilistic (PROB) method to calculate the p\(K_W\)  of water and p\(K_A\)  of an acid, as suggested by Davies et al.,[1]}
based on the work of Chandler.[2]}
This method relies on the relative probability of finding the system in a bound state.
For this purpose we define a cutoff bond distance, \(R_c\) ,
at which the O-H bond breaks and the OH\(^-\)  and H\(_3\) O\(^+\)  ions are formed.
The probability ratio between the bound and dissociated states is given by
\(\gamma (R_c)=\frac{\int _0^{R_c}\exp (-\beta \Delta A(\bar{n}_{\mathrm {O^*}}(r)))4\pi r^2 dr}{\int _0^{R_{\mathrm {max}}}\exp (-\beta \Delta A(\bar{n}_{\mathrm {O^*}}(r)))4\pi r^2 dr},\) 
